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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 2 — Jan. 18, 2010
  • pp: 1380–1388
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Modulation instability in 
high power laser amplifiers

Alexander M. Rubenchik, Sergey K. Turitsyn, and Michail P. Fedoruk  »View Author Affiliations


Optics Express, Vol. 18, Issue 2, pp. 1380-1388 (2010)
http://dx.doi.org/10.1364/OE.18.001380


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Abstract

The modulation instability (MI) is one of the main factors responsible for the degradation of beam quality in high-power laser systems. The so-called B-integral restriction is commonly used as the criteria for MI control in passive optics devices. For amplifiers the adiabatic model, assuming locally the Bespalov-Talanov expression for MI growth, is commonly used to estimate the destructive impact of the instability. We present here the exact solution of MI development in amplifiers. We determine the parameters which control the effect of MI in amplifiers and calculate the MI growth rate as a function of those parameters. The safety range of operational parameters is presented. The results of the exact calculations are compared with the adiabatic model, and the range of validity of the latest is determined. We demonstrate that for practical situations the adiabatic approximation noticeably overestimates MI. The additional margin of laser system design is quantified.

© 2010 OSA

1. Introduction

When the power of a laser beam propagating in nonlinear medium exceeds a critical valuePc, the transverse beam modulations (random or induced) begin to grow exponentially. This physical phenomenon is known as light beam modulation or self-focusing instability [1

1. V. I. Bespalov and V. I. Talanov, “Filamentary Structure of Light Beams in Nonlinear Media,” JETP Lett. 3, 307 (1966).

]. As a result of this instability, the beam quality degrades and the beam breaks into filaments with power of the order ofPc. Every such filament then experiences self-focusing up to the point at which either the high intensity produces the material breakdown, or the field collapse is arrested before breakdown by some other physical effect, depending on the specific configuration and the medium material.

The laser power in modern high-energy pulse laser systems often greatly exceedsPc. For an example, in the National Ignition Facility (NIF) [2

2. C. A. Haynam, P. J. Wegner, J. M. Auerbach, M. W. Bowers, S. N. Dixit, G. V. Erbert, G. M. Heestand, M. A. Henesian, M. R. Hermann, K. S. Jancaitis, K. R. Manes, C. D. Marshall, N. C. Mehta, J. Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C. D. Orth, R. Patterson, R. A. Sacks, M. J. Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C. Widmayer, R. K. White, S. T. Yang, and B. M. Van Wonterghem, “National Ignition Facility laser performance status,” Appl. Opt. 46(16), 3276–3303 (2007). [CrossRef] [PubMed]

] the power is millions of times higher that the critical power. The beam quality in such advanced laser facility is maintained by keeping the cascaded system elements effectively short enough to prevent the dangerous development of the modulation instability [3

3. J. T. Hunt, J. Glaze, W. Simmons, and P. Renard, “Suppression of self-focusing through low-pass spatial filtering and relay imaging,” Appl. Opt. 17(13), 2053 (1978). [CrossRef] [PubMed]

]. Spatial filters are inserted into the system to clean the beam and to prevent the growth of the instability. For passive optical elements (lenses), the self-focusing instability can be treated within linear theory [1

1. V. I. Bespalov and V. I. Talanov, “Filamentary Structure of Light Beams in Nonlinear Media,” JETP Lett. 3, 307 (1966).

]. In amplifiers, where the intensity increases exponentially, an especially careful quantitative treatment is required. Direct numerical modeling of beam propagation can be applied, but this requires a great deal of computational time and is not convenient in multi-parametric system design optimization. Quantitative analytical models are very desirable as they can be used for design analysis before full-scale comprehensive modeling. In this paper, we present exact analytical results providing base for design guidance rules in such complex large-scale laser systems.

2. Physical model and basic equations

We start from the mathematical description of the problem. The nonlinear Schrödinger equation (NSE) governs the propagation of a high-power beam through the amplifier medium according to

iψz+12n0k0Δψ+k0n2|ψ|2ψ=ig02ψ.
(1)

Here k0is the propagation vector in vacuum;n0andn2are the linear and nonlinear refractive indices, respectively; andg0is the amplifier gain. After the straightforward transformation Ψ=exp[g0z/2]×U, this equation reads

iUz+12n0k0ΔU+k0n2(z)|U|2U=0.
(1a)

Heren2(z)=n2(0)exp[g0z], and the optical field propagates from z=0 to z=L. Note that the problem of instability in an amplifier is mathematically similar to the problem of beam propagation in non-uniform media [4

4. A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009). [CrossRef] [PubMed]

] with exponentially increasing nonlinear refractive index. The solution of Eq. (1a) that we are interested in is a plane wave with a z-dependent phase. We consider here a modulation instability of the continuous wave (CW) having the form: U=U0exp[ik0|U0|2n2(z')dz']. The evolution of small perturbations to the CW is given by:

U(z,r)=(U0+a+ib)×exp[ik0|U0|2n2(z')dz',az+12n0k0Δb=0,bz+12n0k0Δa+2k0n2(z)|U0|2a=0.
(2)

These two first-order equations can be combined into the following second-order equation in z:

2az2+12n0k0Δ{12n0k0Δ+2k0n2(z)|U0|2}a=0.

kz2=k22n0k0[k22n0k02k0n2|U0|2].
(3)

It is seen from (3) that the spatial growth kz is increased with k for small values of the latter. It reaches a maximum at
k22n0k0=k0n2|A|2
(4)
and it turns to zero at the point

k22n0k0=2k0n2|A0|2.
(5)

a(0),b(0)exp[kzL]=a(0),b(0)exp[k0n2|A0|2L]=a(0),b(0)exp[B].

Here B is the so-called B integral – a nonlinear phase shift acquired after propagation through the system. To guarantee the beam quality, laser system designers typically require B to be smaller than 2-3. This means that not only perturbations starting from noise, but even induced modulations due to stray light or coating defects, cannot degrade the beam quality. For example, even a perturbation as large as 1% perturbation will not grow to more than 10%.

In the case of a long laser (with B larger than 3), the typical design solution is to insert spatial filters into the system to clean the beam and to reduce effectively the B integral [3

3. J. T. Hunt, J. Glaze, W. Simmons, and P. Renard, “Suppression of self-focusing through low-pass spatial filtering and relay imaging,” Appl. Opt. 17(13), 2053 (1978). [CrossRef] [PubMed]

]. A knowledge of the speed of growth of small perturbations is crucial for design of high power laser facilities (see e.g [2

2. C. A. Haynam, P. J. Wegner, J. M. Auerbach, M. W. Bowers, S. N. Dixit, G. V. Erbert, G. M. Heestand, M. A. Henesian, M. R. Hermann, K. S. Jancaitis, K. R. Manes, C. D. Marshall, N. C. Mehta, J. Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C. D. Orth, R. Patterson, R. A. Sacks, M. J. Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C. Widmayer, R. K. White, S. T. Yang, and B. M. Van Wonterghem, “National Ignition Facility laser performance status,” Appl. Opt. 46(16), 3276–3303 (2007). [CrossRef] [PubMed]

].).

3. The modulation instability in amplifiers

From the design consideration, it is important to be able to identify safe zones within the space of operational parameters. An example would be the amplifier length such that instabilities would not develop to a dangerous level. To start with, one can estimate from the above calculations the perturbation growth due to modulation instability, using the following simple inequality bounding the mode amplitude increase for propagation in the amplifier medium. The evolution of the modes a,bexp[ikr]is given by:

az=k22n0k0b,bz=k22n0k0a+2k0n2(z)|U0|2a.

It can be shown by straightforward manipulations that the z-dependence of the summed amplitudes satisfies |a(z)|2+|b(z)|2exp[2k0|U0|20zn2(z')dz']×(|a(0)|2+|b(0)|2).

This expression gives an exact upper estimate of the perturbation growth. However, a more accurate analysis is desirable for design purpose. Note that in amplifiers, where the initial intensity increases during propagation asI=I0eg0z, the most unstable scale of perturbation decreases during propagation due to the exponential increase of intensity, as seen in (5). A more accurate, and at the same time simple and safe, criteria for good beam quality can be introduced based on the assumption that the maximal growth is given by the mode withkz=k0n2|A(z)|2. Then the corresponding B-integral for an amplifier of length L is defined as in [3

3. J. T. Hunt, J. Glaze, W. Simmons, and P. Renard, “Suppression of self-focusing through low-pass spatial filtering and relay imaging,” Appl. Opt. 17(13), 2053 (1978). [CrossRef] [PubMed]

]:

B'(L)=k0n2(0)0L|U0|2exp[g0z]dz=k0n2(0)|U0|2exp[g0L]1g0=qexp[g0L]14,q=4k0n2(0)|U0|2/g0.
(6)

The typical practical requirement is that B’ is smaller then 3. This is a safe range, but it is still too conservative an estimate, because one assumes that the maximal growth corresponds to the same k2. In reality, during propagation, the higher k2 becomes most unstable. Therefore, it is likely that values of B larger than 3 can be tolerable. We would like to point out that the difference in the accurate estimate of the B-integral is not just an academic interest, but it translates into very practical factors such as, for example, an overall cost of the laser system. Therefore, we further elaborate on different approaches to the finding the safe operational zones, before presenting our results.

A more regular approach based on the so-called adiabatic approximation was used in [5

5. B. R. Suydam, “Self-focusing of very powerful laser beam “ in Laser Induced Damage in Optical Materials, A.J.Glass and A.H.Guenther,Ed.Washington, D,C, NBS,1973,special publication 387,“Self-focusing of very powerful laser beam. II”, IEEE J. Quant. Electron., QE-10 837 (1974)

,6

6. J. Trenholme “LLNL annual report” (1974)

]. One can assume that the transversal perturbation grows faster than the intensity in the amplifier. In particular, this is certainly true for sufficiently high intensities. In this case, one can assume that the perturbations growth follows the intensity adiabatically and (3) is applicable with a z-dependent intensity. We have then the overall amplification factor
γa=0LImkzdz=0.5qf(ω,g0L),ω2=k24n0k02n2(0)|U0|2,
where the function f is defined as

f(ω,g0L)=0g0L[(exω2)ω2]1/2dx:ω2<1f(ω,g0L)=lnω2g0L[(exω2)ω2]1/2dx:ω2>1
(7)

The second expression takes into account that, with the intensity growth, some of initially stable perturbations become unstable. The integrals are straightforward and yield:

f(ω,g0L)=2ω{(eg0Lω2)1/21ω2ωtan1eg0Lω2ω+ωtan11ω2ω};ω2<1f(ω,g0L)=2ω{(eg0Lω2)1/2ωtan1eg0Lω2ω};ω2>1

However, the applicability of the adiabatic approximation is not evident and its accuracy can hardly be controlled a priori. Therefore, in order to determine the safe margin of amplifier operation, we now consider the exact solution of the problem, making corresponding comparison with the results of the adiabatic approach.

In an inhomogeneous medium, the instability evolution is described by Eq. (2), or

d2adz2+k22n0k0[k22n0k02k0n2(z)|U0|2]a=0.
(8)

The initial conditions are: a(0)=f0,dadz|z=0=f1. For n2(z)=n2(0)exp[g0z], introducing ω2=k24n0k02n2(0)|U0|2,ν=qω2=k2n0k0g0,and using q=4k0n2(0)|U0|2/g0, we can write the equation as:

d2adz2+q2ω2g024(ω2exp[g0z])a=0.

After the substitutionx=qωexp[g0z/2], the equation on can be re-written as:

x2d2adx2+xdadx+(ν2x2)a=0.
(9)

A similar equation was derived by M. Karlson [7

7. M. Karlsson, “Modulational instability in lossy optical fibers,” J. Opt. Soc. Am. B 12, 2071 (1995).

] in the context of studies of modulation instability in lossy fibres. We follow here his very important work in many technical aspects. However, apart from an obvious difference in physics, there are also mathematical differences in our analysis compared to [7

7. M. Karlsson, “Modulational instability in lossy optical fibers,” J. Opt. Soc. Am. B 12, 2071 (1995).

]. The key difference is that, while in [7

7. M. Karlsson, “Modulational instability in lossy optical fibers,” J. Opt. Soc. Am. B 12, 2071 (1995).

] only the decaying solution of (8) was considered, here we are interested in the opposite case of a growing solution. The formal solutions of the Eq. (9) are the Bessel functions Iiν(x) and Iiν(x). Therefore the full solution of the initial problem (8) a(z) can be written as:

a(z)=if0qωπ2sinh[νπ]{Iiν'(qω)Iiν(qωeg0z/2)Iiν'(qω)Iiν(qωeg0z/2)}+if1πg0sinh[νπ]{Iiν(qω)Iiν(qωeg0z/2)Iiν(qω)Iiν(qωeg0z/2)}.
(10)

Here the constants are chosen to satisfy the initial conditions: a(0)=f0,dadz|z=0=f1.

The analogue of the increment of the spatial growth rate in our situation will be the valueγ=ln[a(L)/a(0)]. For a large ratio a(L)/a(0), this value is practically independent of boundary conditions. The asymptotic expansion of (10) at large index values coincides with the results of the adiabatic approximation [6

6. J. Trenholme “LLNL annual report” (1974)

]. However, the key issue here is a quantitative analysis. It is not clear a priori in what range of parameters and with what accuracy the adiabatic approximation works and we will discuss this issue in more details below.

The exact solution is the function of three parameters q,g0L andω. The parameter ω2 in the uniform case is the ratio of k2 to its maximal unstable value. For fixed values of gain g0L and an effective nonlinearity q we must find the maximal value of γ as a function ofω. In a uniform media (g0L=0), the most unstable mode corresponds to ω2=1/2 and the cut-off to ω2=1. In the amplifying medium the growing solution does exist in the interval 0ωeg0L/2 and the growth starts when propagation distance is larger than Z*=ln[ω2]/g0. This should be taken into account when calculating increment of instability. The most unstable value of k2 increases in amplifier during the propagation.

We plot in Figs 1–6 solutions of Eq. (10) (show at the point L=1) with the boundary conditions: f0 = 1, f1 = 0 at Z = 0 for 0 < ω ≤ 1, and f0 = 1, f1 = 0 at Z* = ln[ω2]/g0 for 0ωeg0L/2. Figure 1
Fig. 1 γas function of transversal perturbation wave-number ω. Red line q=1 blue q=2 green q=3, black adiabatic approximation and q=1, g0L=3.
shows the function γ(ω) for selected values of q and for g0L=3, while Figure 2
Fig. 2 γ as function of transversal perturbation wave-number ω. Red line g0L=1, blue g0L=2, green g0L=3, black adiabatic approximation g0L=1, q=1.
shows this function for q=1 and for selected values of g0L. The black lines in these graphics correspond to the adiabatic approximation. As expected, the maximum of ω shifts up with increasingg0L. However, for considered solutions the cut-off takes place at ωcut_off=eg0L/2>1 and the most unstable mode corresponds to ω>1. This, in particular, means that the most dangerous modes initially were stable ones and start to grow only later downstream. The result for the adiabatic approximation is plotted for only a single value of q because of the simple q dependence given by (7). It is seen that the adiabatic approximation visibly overestimates the peak value of γ(ω). Note that definition of cut-off frequency is somewhat not well defined as near the cut-off the growth of the mode is not large enough to avoid impact of the initial conditions. An important advantage of the exact analytical solution is that the influence of the initial conditions can be traced directly.

Figure 3
Fig. 3 γ as function q for ω=0.5(red line –exact solution, blue – adiabatic approximation) and for ω=1 (green line – exact solution, black – adiabatic approximation), g0L=3.
shows γ for selected values of ω=0.5,1and g0L, as a function of q, for both the exact solution and the adiabatic approximation. One can see that the curves converge, but only at very high and impractical values of γ. For parameters of practical interest, the adiabatic approximation systematically overestimates the growth of the modulation instability. This is one of the most important practical results of this work. Figure 4
Fig. 4 ωmax as function q for different values of g0L. Red line g0L=3, blue g0L=2, green g0L=1. The dashed lines are results of adiabatic approximation.
shows ω corresponding to the maximal value of γas a function of q, for selected values of g0L.The dashed lines on these pictures represents the results of the adiabatic approximations (7).

As was indicated above, the maximum of ω shifts up with increasingg0L, the cut-off takes place at ω>1, and the most unstable modes corresponds to ω>1. This means that the most dangerous modes initially were stable and started to grow only later downstream. We see that the adiabatic approximation predicts higher values of maxω especially for the high gain situation.

For fixed initial intensity, the increase of q for fixed g0L is equivalent to an increase of the amplifier length for the same final intensity. For higher q, more instability growth takes place at low intensities, which one can see in a decrease of the most unstableω with increasing q. For smaller values of gain, the situation is closer to the uniform case and the shift of ω with change of q is less visible (Fig. 4). In all cases the maximal value of ω is higher then the uniform value1/2.

Figure 5
Fig. 5 Maximum γ (solid line) and B-integral (dashed line) as function of q for different values of g0L. Red line g0L=3, blue g0L=2, green, g0L=1. Dotted lines correspond the adiabatic approximation.
presents the most important result from a practical point of view, which is the instability growth rate as a function of q. For comparison, on the same graph we show the value of the B’ integral given by Eq. (6). The results of the adiabatic approximation also presented here. One can see that the adiabatic approximation noticeably overestimates the growth of the modulation instability. One can see that the real growth is smaller than (6), (7), and that the difference is larger for the high gain situation. For the case g0L=3 and B'=5 the result is the still tolerable value of γ2. The general view ln(γmax(q,g0L)) is shown in Fig. 6
Fig. 6 Counterplot lnγmax as function q and g0L.
.

4. Conclusions

We have presented a theory that accurately describes the growth of the transversal modulations of an optical beam in a laser amplifier. We found the most unstable mode and calculated the growth rate for such mode. In a typical situation, the most unstable modes are stable at the amplifier input and become unstable during the propagation. We demonstrated that the commonly used B’-integral estimate (6), and the more regular adiabatic approximation (7), both substantially overestimate the effects of modulation instability for practically used parameters. The range of applicability of adiabatic approximation is determined. Our results can be used in laser designs to prevent the degradation of beam quality. We also demonstrated that the amplification line for transversal perturbation in the adiabatic approximation is wider in comparison with the exact solution. This means, in particular, that the real system can be less sensitive to perturbations induced by various optics and coatings defects. Our results might also have applications beyond the field of laser science [8

8. H. Segur and D. M. Henderson, “The modulation instability revisited,” Eur. Phys. J. Spec. Top. 147(1), 25–43 (2007). [CrossRef]

,9

9. V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Physica D 238(5), 540–548 (2009). [CrossRef]

].

Acknowledgement

We are grateful to A. Erlandson, K. Manes, and J. Trenholme for useful discussions. This work was partially performed under the auspices of the U. S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344 and interdisciplinary grant 42 from the Siberian Branch of the Russian Academy of Science. The financial support of the EPSRC and the Royal Society is acknowledged.

References and Links

1.

V. I. Bespalov and V. I. Talanov, “Filamentary Structure of Light Beams in Nonlinear Media,” JETP Lett. 3, 307 (1966).

2.

C. A. Haynam, P. J. Wegner, J. M. Auerbach, M. W. Bowers, S. N. Dixit, G. V. Erbert, G. M. Heestand, M. A. Henesian, M. R. Hermann, K. S. Jancaitis, K. R. Manes, C. D. Marshall, N. C. Mehta, J. Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C. D. Orth, R. Patterson, R. A. Sacks, M. J. Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C. Widmayer, R. K. White, S. T. Yang, and B. M. Van Wonterghem, “National Ignition Facility laser performance status,” Appl. Opt. 46(16), 3276–3303 (2007). [CrossRef] [PubMed]

3.

J. T. Hunt, J. Glaze, W. Simmons, and P. Renard, “Suppression of self-focusing through low-pass spatial filtering and relay imaging,” Appl. Opt. 17(13), 2053 (1978). [CrossRef] [PubMed]

4.

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009). [CrossRef] [PubMed]

5.

B. R. Suydam, “Self-focusing of very powerful laser beam “ in Laser Induced Damage in Optical Materials, A.J.Glass and A.H.Guenther,Ed.Washington, D,C, NBS,1973,special publication 387,“Self-focusing of very powerful laser beam. II”, IEEE J. Quant. Electron., QE-10 837 (1974)

6.

J. Trenholme “LLNL annual report” (1974)

7.

M. Karlsson, “Modulational instability in lossy optical fibers,” J. Opt. Soc. Am. B 12, 2071 (1995).

8.

H. Segur and D. M. Henderson, “The modulation instability revisited,” Eur. Phys. J. Spec. Top. 147(1), 25–43 (2007). [CrossRef]

9.

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Physica D 238(5), 540–548 (2009). [CrossRef]

OCIS Codes
(140.0140) Lasers and laser optics : Lasers and laser optics
(140.3280) Lasers and laser optics : Laser amplifiers

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: November 2, 2009
Revised Manuscript: January 4, 2010
Manuscript Accepted: January 4, 2010
Published: January 12, 2010

Citation
Alexander M. Rubenchik, Sergey K. Turitsyn, and Michail P. Fedoruk, "Modulation instability in 
high power laser amplifiers," Opt. Express 18, 1380-1388 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-2-1380


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References

  1. V. I. Bespalov and V. I. Talanov, “Filamentary Structure of Light Beams in Nonlinear Media,” JETP Lett. 3, 307 (1966).
  2. C. A. Haynam, P. J. Wegner, J. M. Auerbach, M. W. Bowers, S. N. Dixit, G. V. Erbert, G. M. Heestand, M. A. Henesian, M. R. Hermann, K. S. Jancaitis, K. R. Manes, C. D. Marshall, N. C. Mehta, J. Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C. D. Orth, R. Patterson, R. A. Sacks, M. J. Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C. Widmayer, R. K. White, S. T. Yang, and B. M. Van Wonterghem, “National Ignition Facility laser performance status,” Appl. Opt. 46(16), 3276–3303 (2007). [CrossRef] [PubMed]
  3. J. T. Hunt, J. Glaze, W. Simmons, and P. Renard, “Suppression of self-focusing through low-pass spatial filtering and relay imaging,” Appl. Opt. 17(13), 2053 (1978). [CrossRef] [PubMed]
  4. A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009). [CrossRef] [PubMed]
  5. B. R. Suydam, “Self-focusing of very powerful laser beam “ in Laser Induced Damage in Optical Materials, A.J.Glass and A.H.Guenther,Ed.Washington, D,C, NBS,1973,special publication 387,“Self-focusing of very powerful laser beam. II”, IEEE J. Quantum Electron. QE-10 837 (1974)
  6. J. Trenholme “LLNL annual report” (1974)
  7. M. Karlsson, “Modulational instability in lossy optical fibers,” J. Opt. Soc. Am. B 12, 2071 (1995).
  8. H. Segur and D. M. Henderson, “The modulation instability revisited,” Eur. Phys. J. Spec. Top. 147(1), 25–43 (2007). [CrossRef]
  9. V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Physica D 238(5), 540–548 (2009). [CrossRef]

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